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Ordered groups
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An ordered group is a group equipped with a linear
ordering
with
for all .
A unary representation of
is an embedding
for
some linearly ordered set
,
where
is the
ordering by universal comparison.
Natural example: itself
with translations on the left, i.e.
.
If is dense or
, then
has QE and is o-minimal.
Two examples
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Each Abelian ordered group embeds in a Hahn ordered group
where is a
linearly ordered set. Each
is represented as a series
.
Let be an ordered
field. Let
be an
ordered vector space over
.
Then
is an ordered
group for the product
and the lexicographic ordering.
We can represent each as
the strictly increasing affine map
,
and the ordering is then
iff
for sufficiently
large
.
Univariate equations
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Fix an o.g. .
: free product of
and
.
Simple task: given
understand the theory of equations
for
lying in extensions
of
.
Remark.
Given
and
, the extension
is determined by an ordering
on the quotient group
where
Examples:
centralizer
extension
extension
with a “square root”.
conjugacy
extension
Abelian and non-Abelian ordered groups
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Assume is Abelian. For
in an Abelian o.g.
extension of
, and
, we have
if and only if
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(1) |
where and
.
There is a universal divisible extension .
The theory of divisible Abelian ordered groups is the model-completion of the theory of Abelian ordered groups. It has QE and is o-minimal.
In the non-Abelian case, solving
in extensions of is
highly non-trivial (recall the functional representation of
...).
Divisibility, conjugacy, problems
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[,
.
[,
.
Question
[
[
Question
Orderability
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An orderable group is a group
which can be ordered.
Every nilpotent torsion-free group is orderable.
Since orderable groups are closed under subgroups, isomorphism and
ultrapowers, they form an elementary class in the first-order
language .
[
So any elementary class of ordered groups in induces an elementary class in
of thus orderable groups.
Miscellany about free groups
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In the 40's,
Both answers are positive [
Free groups can always be ordered [Iwasawa, 1948], and:
The free product of a family of groups can be ordered in such a way as to preserve the ordering on each member of the family.
For every ordered group ,
there is a an ordering on a free-group
and a convex normal subgroup
such that
.
Ordered groups from o-minimal structures
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Let be an o-minimal
structure.
Write for the set of
germs
at
of definable maps
. We have
for a natural “asymptotic” structure on
. We set:
is closed under
composition
, and
is an ordered group.
Example.
:
ordered group.
:
real-closed field.
:
real exponential field.
Example.
:
ordered group.
:
real-closed field.
:
real exponential field.
For , we have
.
Example.
:
ordered group.
:
real-closed field.
:
real exponential field.
For , we have
.
is a divisible
ordered Abelian group, then
.
Example.
:
ordered group.
:
real-closed field.
:
real exponential field.
For , we have
.
is a divisible
ordered Abelian group, then
.
For , we have
. Moreover
has a structure of differential
field [
Example.
:
ordered group.
:
real-closed field.
:
real exponential field.
For , we have
.
is a divisible
ordered Abelian group, then
.
For , we have
. Moreover
has a structure of differential
field [
The field contains
all germs of functions that can be obtained as combinations of
,
, and semialgebraic functions.
A growth axiom
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Let be an ordered
group and let
with
. How is the
inequation
to be solved in ?
In , if
grows much faster than
, then
intuitively grows faster than
.
How “much faster” must
grow? First of all, we should have
for all finite iterations.
More precisely, we should have ,
where
Growth order groups
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Consider the following axioms for an ordered group :
Second axiom: the relation
is an ordering which is compatible with
. The relation
if (
and
) is a convex equivalence relation.
An element is said
central if for all
,
there is a
such that
.
All Abelian ordered groups are growth order groups.
Question
expands the real ordered field, then must
be a growth order group?
Differential fields of series and GOG
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Idea: obtain GOG closed under certain equations
by constructing ordered
differential fields of formal series with composition laws:
Take with
. For
with
, the derivative
and the compositum
are
well-defined.
These operations extend to the ordered field of formal Laurent series.
Problem: The ordered monoid is not a group.
has no inverse
The operations extend to the field
of Puiseux series, and
is a growth order group.
Is divisible?
Are any two
conjugate in
?
Is existentially
closed among growth order groups?
Is divisible?
No, since
has no functional square root.
Are any two
conjugate in
?
No, for instance
and
are not conjugate.
Is existentially
closed among growth order groups? No,
because...
Generalized power series
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Let be a linearly
ordered abelian group. Then
is an ordered field for the Cauchy product
Let be a linearly
ordered abelian group. Then
is an ordered field for the Cauchy product
Certain infinite families can be summed in . We get a “formal Banach
space”:
if is infinitesimal
then
.
we have an implicit function theorem [
equations over with
approximate solutions in
sometime have exact solutions in
Transseries
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Field of
transseries: generalized series involving
,
and
and combinations
thereof.
The number of iterations of
and
must be
uniformly bounded.
enjoys a
derivation
that acts termwise, e.g.
enjoys a
composition law
that acts termwise on the right:
Remark: The elements
and
are conjugate,
via
.
Conjugacy in transseries
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We now look at the sructure .
This is a right-ordered group, i.e for all ,
.
This is a growth order group.
The group is not divisible since
has no square root. There is no “half exponential” which
can be expressed as a combination of exponentials and logarithms.
However:
Any two with
are conjugate.
Since and
has a solution
for all
,
we deduce that
is a divisible growth order group.
Conjucacy and Abel's equation
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Goal: a group
in which
is conjugate
with
.
solve the conjugacy equation
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(2) |
in extensions of .
(2) is the formal version of of
on . Its growth is
transexponential:
for
sufficiently large
.
Question define a transexponential function?
embeds into
.
Can we construct differential fields
with composition where (2) has solutions, and into
which more general
can
be embedded?
Hyperseries
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A differential ordered field with composition where for all
, we have a symbol
with
and
I.e. the inverse equations of (3) are valid. But
is not a group.
An extension
where
is bijective
and strictly increasing.
The derivation
and composition law
on
extend in a
natural way to
.
Conjugacy in hyperseries
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Using Taylor series arguments, one shows that:
The structure is
an ordered group.
In the group , all
conjugacy equations can now be solved:
Any two in
with
are conjugate.
Using this, one shows that:
The structure is
a growth order group.
So we have a GOG solution to our Question 2 on ordered groups.
Surreal numbers
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[ of surreal
numbers. It comes with a linear ordering
of magnitude, and a partial, well-founded
ordering
of
simplicity.
Fundamental property: if are subsets with
,
then there is a unique
-minimal
number
with
.
(And all numbers are constructed in this way.)
Conway inductively defined field arithmetics on . For
,
, we have
The ordered field
naturally contains the reals
as well as the ordinals
with their Hessenberg (commutative) arithmetic.
We have a simplest positive infinite number corresponding to the ordinal
and the series
.
Numbers as hyperseries
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Idea: in ,
asymptotics can be brought to life; for a regular growth rate
and
, define
as the number
We have numbers in
for polynomial asymptotics.
[
with
.
[
on
.
[
and
as in
on surreal numbers.
There is a natural embedding of
into
, and every
surreal number can be represented as a (possibly infinitely
nested) hyperseries in the same vein as elements of
.
There is a natural extension
of the composition law on
to
.
is a growth order
group with exactly three conjucacy classes.
Transfinite non-commutative products
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Let be among
,
,
and
. For
, we have a unique isomorphism
with . Indeed this holds
for
, and is carried over
by conjugacy.
Using the operation, one
can define, for all strictly
-decreasing
ordinal indexed sequence
of “
-simple”
elements of
and sequences
, a transfinite composition
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(4) |
Every element in
can be expressed as in (4), in a unique
way.
In this sense, the GOG
is a non-commutative Hahn product
.
Question
of Abelian ordered groups
indexed by a linearly ordered set
?
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